# Discover prove: Combining Rational and Irrationalnumbers is 1.2+sqrt2 rational or irrational? Is 1/2*sqrt2 rational or irrational? Experiment with sums and products of ther rational and irrational numbers. Prove the followinf. (a) The sum of rational number r and an irrational number t is irrational. (b) The product of a rational number r and an irrational number t is irrational.

Question
Irrational numbers
Discover prove: Combining Rational and Irrationalnumbers is $$\displaystyle{1.2}+\sqrt{{2}}$$ rational or irrational? Is $$\displaystyle\frac{{1}}{{2}}\cdot\sqrt{{2}}$$ rational or irrational? Experiment with sums and products of ther rational and irrational numbers. Prove the followinf.
(a) The sum of rational number r and an irrational number t is irrational.
(b) The product of a rational number r and an irrational number t is irrational.

2021-03-10
Solution:- Combining Rational numbers and irrational numbers:
Rational Numbers:- Numbers which either terminating or none-terminating Repeating. and can be convertible in the form of $$\displaystyle\frac{{p}}{{q}}$$ where $$\displaystyle{p},{q}\in{R}$$ and $$\displaystyle{q}\ne{0}$$ is known as Rational numbers
Irrational numbers:- The Numbers which are non- terminating and non-repeating and all square root of prime numbers is known as Irrational numbers.
Yes $$\displaystyle{12}+\sqrt{{2}}$$ is irrational number.
Yes $$\displaystyle{12}\cdot\sqrt{{2}}=\frac{{1}}{\sqrt{{2}}}$$ is irrational number.
a) prove that sum of rational number and irratoional number is irrational.
Given that :− r be a rational number and t be an irrational number.To prove:− sum of rational number and irratoional number is irrational.
i.e. (r+t)=irrational
Proof:−It is given that r is rational, t is irrational,and assume that is (r+t) rational.
Since a and a+b are rational, we can write them as fraction. Let, r=ap and (r+t)=bp'
$$\displaystyle\Rightarrow{a}{p}+{t}={b}{p}'$$
$$\displaystyle\Rightarrow{t}={b}{p}'−{a}{p}$$
$$\displaystyle\Rightarrow{t}={b}{p}'+{\left(−{a}{p}\right)}$$
$$\displaystyle\Rightarrow{t}=$$ (rational number)+(rational number)
$$\displaystyle\Rightarrow{t}=$$ rational number(sum of two rational numbers is always rational number)
but $$\displaystyle{t}=$$ irratational which is contradiction.
Hence (r+t)= Irrational number.
Hence proved.
b) prove that product of rational number and irratoional number is irrational.
Given that :− r be a rational number and t be an irrational number.
To prove:−product of rational number and irratoional number is irrational.
i.e. (r*t)=irrational
Proof:−It is given that r is rational, t is irrational, and assume that is (r*t) rational.
Since a and a*b are rational, we can write them as fraction
r=ap
and PSK(r×t)=bp'
$$\displaystyle\Rightarrow{a}{p}\cdot{t}={b}{p}'$$
$$\displaystyle\Rightarrow{t}=\frac{{{b}{p}'}}{{{a}{p}}}$$
$$\displaystyle\Rightarrow{t}=\frac{{{b}{p}'}}{{{a}{p}}}$$
$$\displaystyle\Rightarrow{t}=$$ (rational number)/(rational number)
$$\displaystyle\Rightarrow{t}=$$ rational number(rational number of two rational numbers is always rational number)
but t= irratational
Hence (r*t)= Irrational number. Hence proved.

### Relevant Questions

Consider the following statements. Select all that are always true.
The sum of a rational number and a rational number is rational.
The sum of a rational number and an irrational number is irrational.
The sum of an irrational number and an irrational number is irrational.
The product of a rational number and a rational number is rational.
The product of a rational number and an irrational number is irrational.
The product of an irrational number and an irrational number is irrational.
Writing and Proof: If true prove it, if false give a counterexample. Use contradiction when proving.
(a) For each positive real number x, if x is irrational, then $$\displaystyle{x}^{{2}}$$ is irrational.
(b) For every pair of real numbers a nd y, if x+y is irrational, then x if irrational or y is irrational
True or False?
1) Let x and y real numbers. If $$\displaystyle{x}^{{2}}-{5}{x}={y}^{{2}}-{5}{y}$$ and $$\displaystyle{x}\ne{y}$$, then x+y is five.
2) The real number pi can be expressed as a repeating decimal.
3) If an irrational number is divided by a nonzero integer the result is irrational.
The rational numbers are dense in $$\displaystyle\mathbb{R}$$. This means that between any two real numbers a and b with a < b, there exists a rational number q such that a < q < b. Using this fact, establish that the irrational numbers are dense in $$\displaystyle\mathbb{R}$$ as well.
How to find a rational number halfway between any two rational numbers given infraction form , add the two numbers together and divide their sum by 2. Find a rational number halfway between the two fractions in each pair.
$$\frac{1}{4}$$ and $$\frac{3}{4}$$
We need to find a rational number halfway between any two rational numbers given in fraction form, add the two numbers together and divide their sum by 2. Find a rational number halfway between the two fractions in each pair.
$$\frac{1}{100}$$ and $$\frac{1}{10}$$
Given each set of numbers, list the
a) natural Numbers
b) whole numbers
c) integers
d) rational numbers
e) irrational numbers
f) real numbers
$$\displaystyle{\left\lbrace-{6},\sqrt{{23}},{21},{5.62},{0.4},{3}\frac{{2}}{{9}},{0},-\frac{{7}}{{8}},{2.074816}\ldots\right\rbrace}$$
Find
a) a rational number and
b) a irrational number between the given pair.
$$\displaystyle{3}\frac{{1}}{{7}}$$ and $$\displaystyle{3}\frac{{1}}{{6}}$$
In which set(s) of numbers would you find the number $$\displaystyle\sqrt{{80}}$$
1) Find 100 irrational numbers between 0 and $$\displaystyle\frac{{1}}{{100}}.$$